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Theorem atpointN 33308
Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a  |-  A  =  ( Atoms `  K )
ispoint.p  |-  P  =  ( Points `  K )
Assertion
Ref Expression
atpointN  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )

Proof of Theorem atpointN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . . 4  |-  { X }  =  { X }
2 sneq 3978 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
32eqeq2d 2461 . . . . 5  |-  ( x  =  X  ->  ( { X }  =  {
x }  <->  { X }  =  { X } ) )
43rspcev 3150 . . . 4  |-  ( ( X  e.  A  /\  { X }  =  { X } )  ->  E. x  e.  A  { X }  =  { x } )
51, 4mpan2 677 . . 3  |-  ( X  e.  A  ->  E. x  e.  A  { X }  =  { x } )
65adantl 468 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  E. x  e.  A  { X }  =  {
x } )
7 ispoint.a . . . 4  |-  A  =  ( Atoms `  K )
8 ispoint.p . . . 4  |-  P  =  ( Points `  K )
97, 8ispointN 33307 . . 3  |-  ( K  e.  D  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
109adantr 467 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
116, 10mpbird 236 1  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   {csn 3968   ` cfv 5582   Atomscatm 32829   PointscpointsN 33060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-pointsN 33067
This theorem is referenced by: (None)
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